# What is Commutator: Definition and 274 Discussions

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.

View More On Wikipedia.org

5. ### Calculate the following commutator [[AB,iℏ], A]

I've seen this question in a textbook Calculate the following commutator [[AB,iℏ], A] I'm not to sure how you go about it i know [A,B] = AB-BA
6. ### Proving commutator relation between H and raising operator

I am going through my class notes and trying to prove the middle commutator relation, I am ending up with a negative sign in my work. It comes from [a†,a] being invoked during the commutation. I obviously need [a,a†] to appear instead. Why am I getting [a†,a] instead of [a,a†]?
7. ### Can a DC motor work without either a commutator or a controller?

I am a mechanical engineer and my experience with electrical systems is almost nil. The concept of a simple DC motor explained here was quite interesting, especially the need of a commutator part: And then I checked this DIY simple DC motor here and was confused because there was no...
8. ### I Deriving the Commutator of Exchange Operator and Hamiltonian

In the boxed equation, how would you get the right hand side from the left hand side? We know that ##H(1,2) = H(2,1)##, but we first have to apply ##H(1,2)## to ##\psi(1,2)##, and then we would apply ##\hat{P}_{12}##; the result would not be ##H(2,1) \psi(2,1)##. ##\hat{P}_{12}## is the exchange...
9. ### Help with this Commutator question please

Hello, In QM class this morning my Prof claimed that the commutator [𝑥(𝜕/𝜕𝑦), 𝑦(𝜕/𝜕𝑥)] = 0. However, my classmate and I arrived at x(d/dx) - y(d/dy). Can someone explain how (or if) our professor is correct?
10. ### Questions on field operator in QFT and interpretations

For a real scalar field, I have the following expression for the field operator in momentum space. $$\tilde{\phi}(t,\vec{k})=\frac{1}{\sqrt{2\omega}}\left(a_{\vec{k}}e^{-i\omega t}+a^{\dagger}_{-\vec{k}}e^{i\omega t}\right)$$ Why is it that I can discard the phase factors to produce the time...
11. ### Question on discrete commutation relation in QFT

Given the commutation relation $$\left[\phi\left(t,\vec{x}\right),\pi\left(t,\vec{x}'\right)\right]=i\delta^{n-1}\left(\vec{x}-\vec{x}'\right)$$ and define the Fourier transform as...

21. ### Deriving commutator of operators in Lorentz algebra

Li=1/2*∈ijkJjk, Ki=J0i,where J satisfy the Lorentz commutation relation. [Li,Lj]=i/4*∈iab∈jcd(gbcJad-gacJbd-gbdJac+gadJbc) How can I obtain [Li,Lj]=i∈ijkLk from it?

28. ### I Is my reasoning about commutators of vectors right?

Hello guys, I have a question regarding commutators of vector fields and its pushforwards. Let me define a clockwise rotation in the plane \,\phi:\mathbb{R}^2\rightarrow\mathbb{R}^2 \,.\; [\,\partial_x\,,\,\partial_y\,]=0 \,, \;(\phi_{*}\partial_x) = \partial_r and \,(\phi_{*}\partial_y) =...
29. ### I Proving Commutator Identity for Baker-Campbell-Hausdorff Formula

I'm having a little trouble proving the following identity that is used in the derivation of the Baker-Campbell-Hausdorff Formula: $$[e^{tT},S] = -t[S,T]e^{tT}$$ It is assumed that [S,T] commutes with S and T, these being linear operators. I tried opening both sides and comparing terms to no...
30. ### I The Commutator of Vector Fields: Explained & Examples

Hi, I'm just starting to read Wald and I find the notion of the commutator hard to grasp. Is it a computation device or does it have an intuitive geometric meaning? Can anyone give me an example of two non-commutative vector fields? Thanks!
31. ### Commutator group in the center of a group

Homework Statement [G,G] is the commutator group. Let ##H\triangleleft G## such that ##H\cap [G,G]## = {e}. Show that ##H \subseteq Z(G)##. Homework EquationsThe Attempt at a Solution In the previous problem I showed that ##G## is abelian iif ##[G,G] = {e}##. I also showed that...
32. ### A Invariance of Commutator Relations

Does anybody know of examples, in which groups defined by ##[\varphi(X),\varphi(Y)]=[X,Y]## are investigated? The ##X,Y## are vectors of a Lie algebra, so imagine them to be differential operators, or vector fields, or as physicists tend to say: generators. The ##\varphi## are thus linear...
33. ### How to Prove the Commutator Relationship for Angular Momentum Operators?

Homework Statement Show that ##[\hat{L} \cdot \vec{a}, \hat{L} \cdot \vec{b}] = i \hbar \hat{L} \cdot (\vec{a} \times \vec{b})## Homework Equations ##[\hat{L}_i, \hat{L}_j]= i \hbar \epsilon_{ijk} \hat{L}_k ## The Attempt at a Solution [/B] Maybe a naive attempt, but it has been a while. I...
34. ### Angular momentum operator for 2-D harmonic oscillator

1. The problem statement I want to write the angular momentum operator ##L## for a 2-dimensional harmonic oscillator, in terms of its ladder operators, ##a_x##, ##a_y##, ##a_x^\dagger## & ##a_y^\dagger##, and then prove that this commutes with its Hamiltonian. The Attempt at a Solution I get...
35. ### I Can [A,B^n] always equal 0 if [A,B] equals 0?

This is not a homework problem. It was stated in a textbook as trivial but I cannot prove it myself in general. If [A,B]=0 then [A,B^n] = 0 where n is a positive integer. This seems rather intuitive and I can easily see it to be true when I plug in n=2, n=3, n=4, etc. However, I cannot prove it...
36. ### I Commutator of two vector fields

Hello PF, I was reading Carroll’s definition of the commutator of two vector fields in “Spacetime and Geometry”, and I’m having (I think) a simple case of notational confusion. He says for two vector fields, ##X## and ##Y##, their commutator can be defined by its action on a scalar function...
37. ### Delta/metric question (context commutator poincare transf.)

Homework Statement Homework Equations [/B] I believe that ##\frac{\partial x^u}{\partial x^p} =\delta ^u_p ## (1) ##\implies ## (if ##\delta^a_b ## is a tensor, I'm not sure it is?) : ##\frac{\partial x_u}{\partial x^p} = g_{au} \delta ^a_p ## (2) The Attempt at a Solution [/B] sol...
38. ### Paschen back effect and commutator [J^2,Lz]

Homework Statement I have been given a question on how the commutator relates to the paschen back effect the exact question is as follows Calculate the commutator ##[J^2,L_z]## where ##\vec{J}=\vec{L}+\vec{S}## and explain the relevance of this with respect to the paschen back effect Homework...
39. ### A Compute Commutator of Covariant Derivative & D/ds on Vector Fields

Hi, let ##\gamma (\lambda, s)## be a family of geodesics, where ##s## is the parameter and ##\lambda## distinguishes between geodesics. Let furthermore ##Z^\nu = \partial_\lambda \gamma^\nu ## be a vector field and ##\nabla_\alpha Z^\mu := \partial_\alpha Z^\mu + \Gamma^\mu_{\:\: \nu \gamma}...
40. ### A Commutator vector product

Is there some easy way to see that [\vec{p}^2, \vec{p} \times \vec{L}] is equal zero? I use component method and got that.
41. ### A The meaning of the commutator for two operators

Hi, what is the true meaning and usefulness of the commutator in: $$[T, T'] \ne 0$$ and how can it be used to solve a parent ODE? In a book on QM, the commutator of the two operators of the Schrödinger eqn, after factorization, is 1, and this commutation relation...
42. A

### All possible inequivalent Lie algebras

Homework Statement How can you find all inequivalent (non-isomorphic) 2D Lie algebras just by an analysis of the commutator? Homework Equations $$[X,Y] = \alpha X + \beta Y$$ The Attempt at a Solution I considered three cases: ##\alpha = \beta \neq 0, \alpha = 0## or ##\beta = 0, \alpha =...
43. ### I Commutator of p and x/r: Elegant Derivation in Position Basis

This question came up in this thread: <https://www.physicsforums.com/threads/how-to-factorize-the-hydrogen-atom-hamiltonian.933842/#post-5898454> In the course of answering the OP's question, I came across the commutator $$\left[ p_k, \frac{x_k}{r} \right]$$ where ##r = (x_1 + x_2 +...
44. ### Commutator of the Dirac Hamiltonian and gamma 5

Homework Statement Show that in the chiral (massless) limit, Gamma 5 commutes with the Dirac Hamiltonian in the presence of an electromagnetic field. Homework EquationsThe Attempt at a Solution My first question is whether my Dirac Hamiltonian looks correct, I constructed it by separating the...
45. ### A Commutation and Non-Linear Operators

Suppose ##A## is a linear operator ##V\to V## and ##\mathbf{x} \in V##. We define a non-linear operator ##\langle A \rangle## as $$\langle A \rangle\mathbf{x} := <\mathbf{x}, A\mathbf{x}>\mathbf{x}$$ Can we say ## \langle A \rangle A = A\langle A \rangle ##? What about ## \langle A \rangle B =...
46. ### I Problem with Commutator of Gauge Covariant Derivatives?

Hi there, I have just read that the gauge field term Fμν is proportional to the commutator of covariant derivatives [Dμ,Dν]. However, when I try to calculate this commatator, taking the symmetry group to be U(1), I get the following: \left[ { D }_{ \mu },{ D }_{ \nu } \right] =\left( {...
47. ### QFT Klein Gordon Theory, momentum commutator computation

Homework Statement Homework EquationsThe Attempt at a Solution [/B] I think I understand part b) . The idea is to move the operator that annihilates to the RHS via the commutator relation. However I can't seem to get part a. I have: ## [ P^u, P^v]= \int \int \frac{1}{(2\pi)^6} d^3k d^3 k'...
48. ### Quantum Theory, propagator and causality, commutator

Homework Statement Question: To find/ explain why there exists a continuous lorentz transformation that flips the sign for space-like separation but not time-like. Homework Equations Signature ## (-,+,+...) ## Definition of lorentz transformation: ##x^u=\lambda^u_v x^v ##...
49. ### Commutator in the Darwin Term

Homework Statement I am trying to fill in the steps between equations in the derivation of the coordinate representation of the Darwin term of the Dirac Hamiltonian in the Hydrogen Fine Structure section in Shankar's Principles of Quantum Mechanics.  H_D=\frac{1}{8 m^2...
50. ### Angular momentum commutation relations

Homework Statement Show that ##|l, m\rangle## for ##l=1## vanishes for the commutator ##[l_i^2, l_j^2]##. Homework Equations ##L^2 = l_1^2 + l_2^2 + l_3^2## and ##[l_i^2,L^2]=0## The Attempt at a Solution I managed to so far prove that ##[l_1^2, l_2^2] = [l_2^2, l_3^2] = [l_3^2, l_1^2]##. I...